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2E1252 Control Theory and Practice Mikael Johansson [email protected]
2E1252 Control Theory and Practice
Lecture 11:
Actuator saturation and anti wind-up
Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Learning aims
After this lecture, you should • understand how saturation can cause controller states to “wind up” • know how to modify a linear observer to account for saturation • be able to interpret the observer modification in a block diagram • be able to analyze closed-loop stability using the small-gain theorem • know how to tune the anti-windup gain for a PID controller
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So far…
…we have designed linear controllers for linear systems (uncertainty models allow us to account for some nonlinearities)
2E1252 Control Theory and Practice Mikael Johansson [email protected]
G
-F
F r
y
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Actuation is typically limited
In practically all control systems, the actuation is limited The new block represents a saturation
2E1252 Control Theory and Practice Mikael Johansson [email protected]
G
-F
F r
y
u l u
u = sat(ul) =
8><
>:
umin
if ul umin
umax
if ul � umax
ul otherwise
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
The impact of limited actuation
Consider the simple servo model: A controller is which has poles in -13.2444± 13.2255i, and 0.0204 Note: controller is unstable.
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Step response of linear system
A unit step response for the linear closed-loop (assuming no saturation)
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Step response with saturated control
Step responses with actuator saturation for small and large reference change
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What is wrong?
When actuator is saturated, the system operates in open loop (changes in do not affect G or controller states while saturated) Constant input make plant and controller states to grow large (“wind up”)
– particularly critical when plant or controller is unstable or integrating
2E1252 Control Theory and Practice Mikael Johansson [email protected]
G
-F
F r
y
u l u max
u l
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Understanding what is wrong
Consider observer-based controllers (e.g. LQG, H-infinity, H-2, etc.) Controller transfer function: Basic idea of observer: “simulate system” and correct when
– simulation part does not reflect reality when input is saturated!
d
dt
x(t) = Ax(t) +Bul(t) +K(y(t)� y(t))
ul(t) = �Lx(t)
Ul(s) = �Fy(s)Y (s) = �L(sI �A+BL+KC)�1K Y (s)
y(t) 6= y(t)
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An improvement
Make sure that observer reflects actual system dynamics Modification known as observer-based anti-windup
– avoids that controller states wind up – often enough to get reasonable performance
2E1252 Control Theory and Practice Mikael Johansson [email protected]
d
dt
x(t) = Ax(t) +Bu(t) +K(y(t)� y(t))
u(t) = sat(ul(t))
ul(t) = �Lx(t)
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Step responses with modified observer
Step responses with actuator saturation for small and large reference change Smaller overshoot, no longer unstable for large reference changes.
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Analysis: no wind-up in saturation
When in saturation, is constant and controller dynamics is Stability properties given by A-KC, which is typically stable Taking Laplace transforms, we find
d
dt
x(t) = Ax(t) +Bu(t) +K(y(t)� y(t)) =
= (A�KC)x(t) +Bulim +Ky(t)
u(t) = ulim
Ul(s) = �L(sI �A+KC)�1KY (s)�K(sI �A+KC)�1Bulim
s
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Interpretation: feedback from u-ul
To relate modified controller to original, re-write observer as Taking Laplace transforms, we find The linear control law plus compensation from
d
dt
x(t) = Ax(t) +Bu(t) +K(y(t)� y(t)) =
= (A�KC)x(t) +B(ul(t) + u(t)� ul(t)) +K(y(t)� y(t)) =
= (A�BL�KC)x(t) +Ky(t) +B(u(t)� ul(t))
Ul(s) = �L(sI �A+BL+KC)�1KY (s)
� L(sI �A+BL+KC)�1B(U(s)� Ul(s)) =
= �Fy(s)Y (s) +W (s)(U(s)� Ul(s))
u(t)� ul(t)
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Interpretation in block diagram
Basis for many heuristic techniques for anti-windup (more later…)
G
-F
F r
y
u l u
W
-
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What about stability?
We have shown that – in absence of saturation, the closed-loop system is stable – when input remains in saturation, the controller is stable
but no stability guarantees when control moves in and out of saturation! Global stability can sometimes be ensured using small-gain theorem
2E1252 Control Theory and Practice Mikael Johansson [email protected]
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A small-gain analysis
To find the linear system M, note that Since gain of saturation nonlinearity is one (cf. Lecture 1), so if closed-loop stability is ensured
2E1252 Control Theory and Practice Mikael Johansson [email protected]
G
-F
F r
y
u l u
W
-
u l u
M
ul = W (u� ul)�GFyu ) (I +W )ul = (W �GFy)u := Mu
kMk1 < 1
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Design guidelines
1. Design observer-based controller using technique of choice 2. Modify observer to reflect the presence of saturation nonlinearity 3. Attempt to establish stability using small gain theorem, simulate 4. If unsatisfactory, re-design controller with higher penalty on control
2E1252 Control Theory and Practice Mikael Johansson [email protected]
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Application to PID control
Common choice: (tracking anti-reset windup) Rule-of thumb: (Ti integral time, Td derivative time of PID)
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2E1252 Control Theory and Practice Mikael Johansson [email protected]
Summary
The problem with limited actuation: • New phenomena, not predicted by linear control theory
– controller, plant states “wind up” (grow large), feedback loop open • Observer-based anti-windup
– make estimator reflect actual process dynamics – ensures that controller states are stable in saturation – interpretation as feedback from “saturation error” – global stability analysis via small-gain theorem
• Anti-windup for PID control – same structure, heuristic compensator, rules-of-thumbs